Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods

K M Saad; M M Khader; J F Gómez-Aguilar; Dumitru Baleanu

doi:10.1063/1.5086771

Numerical solutions of the fractional Fisher's type equations with Atangana-Baleanu fractional derivative by using spectral collocation methods

Chaos. 2019 Feb;29(2):023116. doi: 10.1063/1.5086771.

Authors

K M Saad¹, M M Khader², J F Gómez-Aguilar³, Dumitru Baleanu⁴

Affiliations

¹ Department of Mathematics, Faculty of Arts and Sciences, Najran University, P.O. Box 1988, Najran City, Saudi Arabia.
² Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 1440, Riyadh, Saudi Arabia.
³ CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, 62490 Cuernavaca, Morelos, Mexico.
⁴ Department of Mathematics, Faculty of Sciences, Cankaya University, Balgat, P.O. Box 06530, Ankara, Turkey.

PMID: 30823705
DOI: 10.1063/1.5086771

Abstract

The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher's equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.